Proposition 8 [1]} 
If \(w\in H^1(\mathbb {R})\) , then
\(\sup _{x \in \mathbb {R}}\left\Vert \int _{-\infty }^{x} \mathrm {e}^{2 i z(x-y)} w(y) dy\right\Vert _{L_{z}^2(\mathbb {R})} \le \sqrt{\pi }\Vert w\Vert _{L^2} ,\) 
